It feels great contributing to Visual Insight images!

It feels great contributing to Visual Insight images!

Originally shared by John Baez

{7,3,3} meets the plane at infinity

Maryam Mirzakhani won the Fields medal largely for her work on hyperbolic geometry.  The hyperbolic plane is something we can all enjoy: it's not flat, but curved like a saddle, and triangles have angles that add up to less than 180 degrees.  We can draw it crushed down to a disk, as in this picture by Roice Nelson.  All the light blue circles should really be the same size - but the ones near the edge have been squashed.

Instead of learning Euclidean geometry in school, you could have learned hyperbolic geometry - it's almost the same, with some big differences.  The main thing is that the parallel postulate is false in hyperbolic geometry: there are many ways to draw a line through a point parallel to another line!  This is how Lobachevsky  discovered hyperbolic geometry in 1823: by seeing what would happen if you changed the rules this way.

And instead of learning trigonometry, you could have learned hyperbolic trigonometry - it's very similar, but you have identities like

cosh²θ - sinh²θ = 1

The most exciting thing is that you could do in this alternate universe is take the hyperbolic plane, cut out carefully chosen pieces, and fold it into multi-holed doughnut shapes without wrinkling the paper.  

These shapes were intensively studied by Felix Klein and Henri Poincaré... and Maryam Mirzakhani is carrying on this grand tradition.  They're important in number theory, string theory and many other subjects.

Hyperbolic geometry works in higher dimensions, too.  In my last Visual Insight post, I showed you a 3-dimensional 'hyperbolic honeycomb', a marvelous pattern built of sheets of regular heptagons.  It's called the {7,3,3} honeycomb because each heptagon has 7 sides, the heptagons meet in groups of 3 on each sheet, and 3 sheets meet along each edge of each heptagon.

This is a view of the 'sky' in a 3-dimensional world with a {7,3,3} honeycomb in it.  To understand it, check out my latest Visual Insight post:

http://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/

and compare the picture there with the previous one:

http://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/

#geometry